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Bloch Oscillations

Bloch Oscillations. Alan Wu April 14, 2009 Physics 138. Outline. Phenomenon Description Semi-Classical Derivation Wannier-Stark States Implications and Applications Terahertz Oscillations Bloch Oscillation Transistors. Bloch Oscillation Phenomenon. Described by Bloch (1928)

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Bloch Oscillations

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  1. Bloch Oscillations Alan Wu April 14, 2009 Physics 138

  2. Outline • Phenomenon Description • Semi-Classical Derivation • Wannier-Stark States • Implications and Applications • Terahertz Oscillations • Bloch Oscillation Transistors

  3. Bloch Oscillation Phenomenon • Described by Bloch (1928) • Imagine a particle in a periodic potential acted on by a constant force. • Example: electrons in crystal lattice exposed to constant electric field • Classically, we expect Ohmic behavior

  4. Bloch Oscillation Phenomenon • But quantum mechanics predicts that the particle will undergo an oscillation • The periodicity causes the group velocity of the wavefunction to oscillate • Ohmic behavior results from scattering

  5. Bloch Oscillation Frequency • Use invariance: shift 1 period d and shift energy ΔE • Phase shift now • Corresponding frequency is

  6. Semi-Classical Derivation • Schrodinger’s Equation can be transformed into the form: • Known as the Acceleration Theorem, since it describes change in momentum • Like classical relation between momentum and force

  7. K-Space in a Lattice • Potential periodicity in real space => periodicity in k-space • Also known as a reciprocal lattice

  8. The Brillouin Zone • Brillouin Zone: a basic cell in the reciprocal lattice • The dispersion relation gives an oscillating k within this zone

  9. Kronig Penney Model • The Kronig Penney model for a lattice can be used to find the potential in k-space. Source: http://fermi.la.asu.edu/schmidt/applets/kp/plugkp.html

  10. Dispersion in lattice

  11. Wannier-Stark Resonance States • At each well, a series of energies are available, much like that of a harmonic resonator. • These states form what is known as a Wannier-Stark energy ladder.

  12. Tight-binding model • Consider just interactions between neighboring wells (known as Wannier representation) • Also have energy difference from constant force

  13. Experimental Confirmations • Bloch oscillations have been observed in semiconductor lattices • Shining a laser will excite the Wannier Stark states, which then oscillate. • These oscillations can be measured

  14. Terahertz Radiation • Changing the electric field allows for a tunable radiation source. • Can get frequencies in the terahertz

  15. Bloch Oscillation Transistors • Bloch oscillations can control Josephson Junctions • Act much like bipolar transistors

  16. Conclusion • Bloch oscillations are just another strange quantum phenomenon • They can be used for frequencies in the terahertz range • Bloch oscillator transistors are an interesting way of amplifying signals

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